Monsky's theorem
In geometry, Monsky's theorem states that it is not possible to dissect a square into an odd number of triangles of equal area.[1] In other words, a square does not have an odd equidissection.
The problem was posed by Fred Richman in the American Mathematical Monthly in 1965, and was proved by Paul Monsky in 1970.[2][3][4]
Proof
Monsky's proof combines combinatorial and algebraic techniques, and in outline is as follows:
- Take the square to be the unit square with vertices at (0,0), (0,1), (1,0) and (1,1). If there is a dissection into n triangles of equal area then the area of each triangle is 1/n.
- Colour each point in the square with one of three colours, depending on the 2-adic valuation of its co-ordinates.
- Show that a straight line can contain points of only two colours.
- Use Sperner's lemma to show that every triangulation of the square into triangles meeting edge-to-edge must contain at least one triangle whose vertices have three different colours.
- Conclude from the straight-line property that a tricolored triangle must also exist in every dissection of the square into triangles, not necessarily meeting edge-to-edge.
- Use Cartesian geometry to show that the 2-adic valuation of the area of a triangle whose vertices have three different colours is greater than 1. So every dissection of the square into triangles must contain at least one triangle whose area has a 2-adic valuation greater than 1.
- If n is odd then the 2-adic valuation of 1/n is 1, so it is impossible to dissect the square into triangles all of which have area 1/n.[5]
Generalisations
The theorem can be generalised to higher dimensions: an n-dimensional hypercube can only be divided into simplices of equal volume if the number of simplices is a multiple of n!.[2]
References
- ^ Aigner, Martin; Ziegler, Günter M. (2010). "One square and an odd number of triangles". Proofs from The Book (4th ed.). Berlin: Springer-Verlag. pp. 131–138. doi:10.1007/978-3-642-00856-6_20.
- ^ a b Sperner's Lemma, Moor Xu
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|doi=10.1007/BF02985395instead. - ^ Dissecting a square into triangles